In this work we explored a generalization of the study of adiabatic
cycles initiated in the context of quantum mechanics by
Berry [1]}. A geometric phase becomes non-trivial if the
adiabatic cycle over which it is defined encircles a degeneacy point.
If the path is modified so that it crosses the degeneracy point, the
Berry phase becomes undefined. The direction of our generalization
was the consideration of cycles which cross isolated degeneracy
points. We showed that it is possible to define quantities which are
physically well defined in this case. The quantities which achieve
the above goal are ratios of the cumulants associated with adiabatic
cycles. We gave a detailed analysis of how cumulant generating
functions can be defined in the general context of the geometric
phase, as well as the particular context of the modern theory of
polarization. For the latter we drew on the work of Souza, Wilkens,
and Martin  [2]}.
